Heat conduction in 1D harmonic crystal: Discrete and continuum approaches

Abstract

In this work the energy transfer in a one-dimensional harmonic crystal is investigated. In particular, a comparison between the discrete approach presented by Klein, Prigogine, and Hemmer with the continuum approach presented by Krivtsov is made. In the pioneering work of Klein and Prigogine the transfer of thermal energy is considered. In particular, an expression is obtained, which allows to calculate the thermal energy of each particle as a function of time. Later, Hemmer derived and used similar expressions to solve several particular problems in context of heat conduction. In the work of Krivtsov—in contrast to the discrete approach—a partial differential continuum equation is derived from the lattice dynamics of a 1D harmonic crystal. This so-called ballistic heat equation describes the propagation of heat at a finite speed in a continuous one-dimensional medium. The current work compares analyses based on the discrete equation of Klein, Prigogine, and Hemmer with those from the continuum-PDE-based one by Krivtsov. There is an important difference between the approaches. The continuum approach is derived from the dynamics of the crystal lattice, in which only kinetic degrees of freedom were excited and then thermal equilibration occurred. In contrast to that we consider in the discrete approach explicitly given equal kinetic and potential initial energies. Several exactly solvable initial problems are studied by using both methods. The problem of point perturbation shows a discrepancy in the results obtained in the framework of the continuous and discrete approaches. It is caused by the fact that the smoothness conditions of the initial perturbation is violated for the continuum approach. For other problems it is shown that at large spatial scales, where the one-dimensional crystal can be considered as a continuous medium, the discrete and the continuum relations converge. The asymptotic behavior of the difference between two aforementioned approaches is analyzed.